We are given the following integrals:

$\int_a^b f(x) \, dx = 9 \quad \text{and} \quad \int_a^b g(x) \, dx = -2$

We need to find:

$\int_a^b \left( 2f(x) - g(x) \right) dx$

Using the linearity property of integrals, we can break this down as follows:

$\int_a^b \left( 2f(x) - g(x) \right) dx = \int_a^b 2f(x) \, dx - \int_a^b g(x) \, dx$

Now, using the fact that the integral of a constant times a function can be factored as:

$\int_a^b 2f(x) \, dx = 2 \int_a^b f(x) \, dx = 2 \times 9 = 18$

Therefore, we have:

$\int_a^b \left( 2f(x) - g(x) \right) dx = 18 - (-2) = 18 + 2 = 20$

Thus, the value of the integral is:

$\int_a^b \left( 2f(x) - g(x) \right) dx = 20$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the result change if $\int_a^b g(x) dx$ was positive instead of negative?
2. Can you generalize the result for $\int_a^b \left( cf(x) + dg(x) \right) dx$ where $c$ and $d$ are constants?
3. What is the geometric interpretation of adding two integrals like this?
4. How does the linearity of integrals simplify complex integrals in general?
5. How would the solution change if we integrated from $b$ to $a$ instead?

Tip: Always utilize the linearity of integrals when dealing with sums and scalar multiples of functions!